COMPUTATION OF LOGISTIC LAW OF GROWTH

QUICK METHOD FOR COMPUTATION OF LOGISTIC LAW OF GROWTH IN BIOLOGY

RAJENDRA DHOBAL1 SAHIL PERSHAD2

Introduction:

Rapid advances are being made in the application of cell and tissue culture techniques for the purpose of genetic manipulation in plants. Also the various inputs control the growth of cells in a variety of manners and studies in these fields are done using various segregations of a population under changing environments (Cocking, 1978). Many a times the growth processes in biological and botanical fields are governed both by internal and external factors. The intake of some nutrients control this process in an entirely internal way while temperatures, solar energy, photo interactions govern them externally.

 

These processes of growth have been dealt mathematically using basic assumption and a natural law – called Logistic law of growth have been put forward by volterra (1920), Feller (1940), Killon & Grant (1995) and many such attempts have been made. Ecologists and field biologist have been interested in the impact of invading species on native biotas for a long time (Elton, 1958). The process of colonization and geographical spread of invading species currently is a tropic of considerable interest in ecology (Mooney et.al., 1986, Drabe and Mooney, 1989) and practical interest in natural resource management (Killon and Grant, 1993). Historically, diffusion models most commonly have been used to describe spatial patterns of colonization of invading species (Skellam, 1951).

To make accurate predictions regarding ultimate range of S. Invicta etc. a better understanding of the dynamics of individual colonies is needed and in this work a quick method is proposed to predict the colonial distribution with time as the colony must be distributed to determine the total number of individual’s present y (t) at any time is difficult to measure.

Formulation and solution of the problem:

Let Y(t) be the population of a bacterial colony at a time t , then the growth of this colony with ø time can be put as follow

bb1

Where function f(y) is dependent upon the population at a time. This will be dependent upon environmental, mutual interaction of bacteria (predator-prey like interactions) etc. The origin of time – measuring is arbitrary because this always can be made convenient to one’s observations.

As a general theory, the function y(t) can be therefore expressed as

bb2

We shall assume that

  1. The growth rate of bacterial population is proportional to the population at time t;
  2. The growth rate is such that this has a maximum limit α and therefore it is also proportional to (α – y(t) ) i.e. the growth rate is proportional to residual (α – y(t) ).

This is in congruence with practical situations as we know from experience that the population of any bacteria doesn’t go out infinitely and has a finite limit and shall be controlled by ecological conditions in the environment. Therefore we come the growth law as

bb3

Here β, α are constants and dependent on factors described above. When we integrate this law, we get

bb4 ------------- (1)

Where m is some reference time.

This law has been useful in predicting population growth of countries, biological populations, bacterial colonies, fruit flies, growth of rats, growth of sunflowers, and many phenomena concerned with growth and is useful to compute the invading populations.

Methology of Application & Computation:

Experimentally, the populations y(t) are only recorded under some environmental conditions such as temperature, pollution, ingredients etc. We shall fix these factors as a whole instantly i.e. factors β, α, m are assumed constants else they are functions of these factors. (Model Figure 1).

Let us make observations Yi at regular intervals of time ti . Then for any ith observation we can write (1) as follows:

bb5 --------------- (2)

By some rearrangements we can write (2) as follows:

Zi+1 = m’Zi --------------- (3)

Following relations can be established

Zi+1 = mZi --------------- (4)

Zi+2 = m Zi+1 and so on.

Using experimental data Z’s, we can find a best statistical value by minizing

bb6

Where S = X’s value or objective function.

bb7 Provides the best estimate of m and therefore

bb8 --------------------------- (6)

Using this value of bb19, we again proceed as follows:

bb9 ---------------------------(7)

From which we estimate in as follows:

bb10 = bb11 = bb12 ---------------------------(8)

and from which we get

bb13 ---------------------------- (9)

Therefore a table of Zi, Zi+1/Zi, k,m has to be made and the growth law is calculated on simple calculator & useful to field scientists.

APPLICATIONS

Table 1

 

Area in square Centimeters

Age

Days

0

1

2

3

4

5

Observed

0.24

2.78

13.53

36.30

47.50

49.40

Lotka & Kostzin

0.25

2.03

13.08

37.05

47.39

49.02

Theoretical Method

0.37

2.51

13.62

36.30

47.42

49.55

Present method

Present calculation method gives

α = 49.40, 49.92 (theoretical)

bb19 = 2.115, 1.95 (theoretical)

bb20 =2.4659, 2.5(theoretical)

Comparison with best results is shown above and it is quite accurate (Figure2).

bb14

(Figure 2 )

The theoretical modal is

bb15

While present model is

bb16.

References:

Cocking, E.C., 1978, Protoplast culture and somatic hybridization In: Proceedings of symposium on plant Tissue culture. Science Press, Peking, p.255-263.

Volterra, V, 1926, Fluctuations in abundance of a species considered mathematically, Nature, 118, p.558-560.

Feller,W, 1940, On logistic law of growth and its empirical verification in Biology, in Mathematics & Application, Ellis Horwod series.

Killon,M.J. and Grant,W.E., 1995, A colony growth model for imported fire ant: potential geographic range of an invading species, Ecological modeling,77, p.73-84.

Killon,M.J. and Grant,W.E., 1993, Scale effects in assessing the impact of imported fire ants on small mammals., Southwest Nat.38, p.393-396.

Elton, C.S., 1958, The ecology of invasions by animals and plants, Methuen, London, 181 p.

Money, H.A., Drake, J.A. and Baker, H.G., 1986, Ecology of Biological invasions of N.America and Hawai,Springer-Verlay, N.Y., 321 p.

Skellam, J.G., 1951, Random dispersal in theoretical population, Biometrika, 38, p.196-218.

Lofgren,C.S., 1986, History of imported fire ants in USA In: C.S. Lofgren and R.K.Vander Meer ( Editors), Westview Press, Boulder, CO. p.36-47.

Model

bb17

Figure 1:- Some factors controlling the growth